Question

Determine De-Broglie Wave equation.

Answer 1

De-Broglie Wave: The waves which are associated with matter are called matter waves or de Broglie waves.
Wave equation: According to quantum theory, the energy of a photon is given by the formula:
$E=hv$ .....(i)
where, $h=$ Planck constant and $v=$ Frequency of light.
Suppose the mass of photon is $m$.
$\therefore$ By Einstein's mass energy relation,
$E=m{c}^2$ ......(ii)
where, $c=$ velocity of light.
Now, from equation (i) and (ii), we get
$m{c}^2=hv\Rightarrow m=\frac{hv}{c^2}$
But, $v=\frac{c}{\lambda }$ $\therefore m=\frac{hc}{\lambda c}$
$m=\frac{h}{\lambda c}$ .....(iii)
If the momentum of photon be $p$, then
$p=mc$
or $p=\frac{hc}{\lambda c}$ [from equation (iii)]
or $p=\frac{h}{\lambda }\Rightarrow \lambda =\frac{h}{p}$
Which is called De-Broglie wave equation.
Again, if the velocity of mass $m$ is $v$, then
$p=mv$ and $\lambda =\frac{h}{mv}$
which is De-Broglie relation.